Roster and Set-Builder

A set can be written two ways: by listing its members (roster form), or by stating the rule they all obey (set-builder form). Pick a rule and watch both forms appear together, with every number from 1 to 20 lighting up if it belongs.

Set-builder form (the rule)

Roster form (the list)

Each rule is one set, shown in both forms Highlighted numbers satisfy the rule The two forms describe the same set

Reading the rule

The notation

Set-builder form reads: { x : x satisfies some rule }. The colon means “such that”. So { x : x is even, x < 10 } is read “the set of all x such that x is even and less than 10” — which lists as {2, 4, 6, 8}.

When to use each

Roster form is clearest for small sets you can simply list. Set-builder form is essential for large or infinite sets — you cannot list all even numbers, but “{ x : x is even }” captures them in one line.